复椭圆对称分布的半参数CRB和Slepian-Bangs公式
Semiparametric CRB and Slepian-Bangs formulas for Complex Elliptically Symmetric Distributions
摘要：
本文的主要目的是扩展半参数推断方法，最近调查的实椭圆对称（RES）分布，复椭圆对称（CES）分布。在利用所采集数据的复表示的所有实际应用中，向复域的推广是至关重要的。此外，CES分布已被广泛认为是一种有价值的通用模型，用于统计描述源自各种物理测量过程的数据集的非高斯行为。本文分为两部分。第一部分利用\textit{Wirtinger}或$\mathbb{C}\mathbb{R}$ - \textit{calculus}，得到了一组CES分布随机向量的复均值向量和复散布矩阵联合估计的约束半参数Cramér-Rao界（CSCRB）的封闭表达式。第二部分论述了在CES模型的背景下Slepian-Bangs公式的半参数版本的推导。具体而言，建议的半参数Slepian-Bangs（SSB）公式为我们提供了一个有用的和现成的表达的半参数Fisher信息矩阵（SSIM）的参数向量的估计参数化的复杂的平均值和复杂的散布矩阵的CES分布的矢量中存在未知的，滋扰，密度发生器。此外，我们展示了如何利用推导出的SSB公式，以获得半参数对应的随机CRB的波达方向（DOA）估计下的随机信号模型的假设。仿真结果也提供澄清的理论研究结果，并证明其实用性，在常见的阵列处理应用。
英文摘要：
The main aim of this paper is to extend the semiparametric inference methodology, recently investigated for Real Elliptically Symmetric (RES) distributions, to Complex Elliptically Symmetric (CES) distributions. The generalization to the complex field is of fundamental importance in all practical applications that exploit the complex representation of the acquired data. Moreover, the CES distributions has been widely recognized as a valuable and general model to statistically describe the non-Gaussian behaviour of datasets originated from a wide variety of physical measurement processes. The paper is divided in two parts. In the first part, a closed form expression of the constrained Semiparametric Cramér-Rao Bound (CSCRB) for the joint estimation of complex mean vector and complex scatter matrix of a set of CES-distributed random vectors is obtained by exploiting the so-called \textit{Wirtinger} or $\mathbb{C}\mathbb{R}$-\textit{calculus}. The second part deals with the derivation of the semiparametric version of the Slepian-Bangs formula in the context of the CES model. Specifically, the proposed Semiparametric Slepian-Bangs (SSB) formula provides us with a useful and ready-to-use expression of the Semiparametric Fisher Information Matrix (SFIM) for the estimation of a parameter vector parametrizing the complex mean and the complex scatter matrix of a CES-distributed vector in the presence of unknown, nuisance, density generator. Furthermore, we show how to exploit the derived SSB formula to obtain the semiparametric counterpart of the Stochastic CRB for Direction of Arrival (DOA) estimation under a random signal model assumption. Simulation results are also provided to clarify the theoretical findings and to demonstrate their usefulness in common array processing applications.
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